Tiles and Electrical Heated floor mats on Concrete floor.?

You will need to use something as a thermal break between the slab and your slate. The electric heating mats will run constantly trying to heat the basement slab.

Putting down backer board is a waste of time and money when going over a smooth cement floor. Backer board is used to stiffen up an existing wooden floor or wall. It can also be used when planning a shower to allow for build up to slope to a drain. As far as sweating is concerned backer board will not prevent it if it is a problem. It will just make your floor thicker. The sweat comes from the air contact with a cooler floor surface it is not from water seeping through the floor. A dehumidifier will help to solve this problem in the summer months. Make sure you use a sufficient base of thinset (1/4 inch trowel minimum, 1/2 inch if uneven or rough spots). Mixing a latex additive to the grout is also a good idea in below grade and outdoor grouting situations.

In addition to losing a lot of heat into the ground, subfloor heating systems cost a lot more than conventional heat and fail more than the vendors admit. It is the more expensive way to go. On the + side, it is nice to have warm feet and not see heaters along the walls. If you have the option, use gas forced air heat instead of electric.

at least your thinking why not make it easy and go with baseboard